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The Universal Model for the negation-free fragment of IPC

Apostolos Tzimoulis and Zhiguang Zhao November 21, 2012

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

Universal Model

The n-universal model for IPC, U(n) = (U(n), R, V ) is the “least” model of IPC that witnesses the failure of every unprovable formula of IPC.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

Universal Model

The first layer U(n)1 consists of 2n nodes with the 2n different n-colors under the discrete ordering. Under each element w in U(n)k \ U(n)k−1, for each color s < col(w), we put a new node v in U(n)k+1 such that v ≺ w with col(v) = s, and we take the reflexive transitive closure of the ordering. Under any finite anti-chain X with at least one element in U(n)k \ U(n)k−1 and any color s with s ≤ col(w) for all w ∈ X, we put a new element v in U(n)k+1 such that col(v) = s and v ≺ X and we take the reflexive transitive closure of the ordering. The whole model U(n) is the union of its layers.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

Example: n = 1

The Rieger-Nishimura ladder:

• 1

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

Properties of U(n)

Lemma For any finite rooted Kripke n-model M, there exists a unique w ∈ U(n) and a p-morphism of M onto U(n)w. Theorem For any n-formula ϕ, U(n) | = ϕ iff ⊢IPC ϕ.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

de Jongh formulas for U(n)

Proposition For every w ∈ U(n) we have that V (ϕw) = R(w), where R(w) = {w′ ∈ U(n)|wRw′}; V (ψw) = U(n)\R−1(w), where R−1(w) = {w′ ∈ U(n)|w′Rw}.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

de Jongh formulas for U(n)

For any node w in an n-model M, if w ≺ {w1, . . . , wm}, then we let prop(w) := {pi|w | = pi, 1 ≤ i ≤ n}, notprop(w) := {qi|w qi, 1 ≤ i ≤ n}, newprop(w) := {rj|w rj and wi rj for each 1 ≤ i ≤ m, for 1 ≤ j ≤ n}.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

de Jongh formulas for U(n)

If d(w) = 1, then let ϕw := prop(w) ∧ {¬pk|pk ∈ notprop(w), 1 ≤ k ≤ n}, and ψw := ¬ϕw. If d(w) > 1, and {w1, . . . , wm} is the set of all immediate successors of w, then define ϕw := prop(w) ∧ ( newprop(w) ∨

m

• i=1

ψwi →

m

• i=1

ϕwi), and ψw := ϕw →

m

• i=1

ϕwi.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

Universal Model and Henkin Model

Lemma For any w ∈ U(n), let ϕw be the de Jongh formula of w, then we have that H(n)Cn(ϕw) ∼ = U(n)w. Lemma Upper(H(n)) is isomorphic to U(n).

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Preliminaries

The top model property and negation-free formulas

Definition (Top-Model Property) ϕ has the top-model property (TMP), if for all M, w, M, w | = ϕ iff M+, w | = ϕ, where M+ is obtained by adding a top point t such that all proposition letters are true in t. Proposition

1 If ϕ ∈ [∨, ∧, →] then it has the TMP, and so has ⊥. 2 For any formula ϕ, there exists a formula ϕ∗ ∈ [∨, ∧, →] or

ϕ∗ =⊥ such that for any top-model (M+, w), (M+, w) | = ϕ ↔ ϕ∗.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Definitions

Universal Model for [∨, ∧, →]-fragment

The n-universal model for the negation-free fragment of IPC, U⋆(n) = (U⋆(n), R⋆, V ⋆), is a generated submodel of the universal model for IPC. It is (generated by): {u ∈ U(n) : ¬uRw0} where w0 is the maximal element of U(n) that satisfies all propositional atoms.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Definitions

Universal Model for [∨, ∧, →]-fragment

The first layer U⋆(n)1 consists of 2n − 1 nodes with all the different n-colors – excluding the color 1 . . . 1 – under the discrete ordering. Under each element w in U⋆(n)k \ U⋆(n)k−1, for each color s < col(w), we put a new node v in U⋆(n)k+1 such that v ≺ w with col(v) = s, and we take the reflexive transitive closure of the ordering. Under any finite anti-chain X with at least one element in U⋆(n)k \ U⋆(n)k−1 and any color s with s ≤ col(w) for all w ∈ X, we put a new element v in U⋆(n)k+1 such that col(v) = s and v ≺ X and we take the reflexive transitive closure of the ordering. The whole model U⋆(n) is the union of its layers.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Definitions

Examples

The 1-universal model is a singular point:

For n ≥ 2 it is infinite.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Definitions

Positive morphisms

Definition A positive morphism is a partial function f : (W , R, V ) → (W ′, R′, V ′) such that:

1 dom(f ) ⊇ {w ∈ W : ∃p ∈ Prop(w /

∈ V (p))}.

2 If w, v ∈ dom(f ) and wRv then f (w)R′f (v). 3 If w ∈ dom(f ) and f (w)R′v then there exists some

u ∈ dom(f ) such that f (u) = v and wRu (back).

4 If w ∈ dom(f ) and vRw, then v ∈ dom(f ) (downwards

closed).

5 For every p ∈ Prop we have w ∈ V (p) ⇐

⇒ f (w) ∈ V ′(p). If the models are descriptive we furthermore require for every Q ∈ Q that W \ R−1(f −1[W ′ \ Q]) ∈ P. These maps restrict strong partial Esakia morphisms.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Definitions

Strong positive partial Esakia morphisms

Lemma Let f : (W , R, V ) → (W ′, R′, V ′) be a positive morphism. Then for every ϕ ∈ [∨, ∧, →] and w ∈ dom(f ) we have that (W , R, V ), w | = ϕ if and only if (W ′, R′, V ′), f (w) | = ϕ. Proof. If (W ′, R′, V ′), f (w) | = ϕ → ψ then if (W , R, V ), v | = ϕ with wRv, then either v ∈ dom(f ) and we use the induction hypothesis,

• r v /

∈ dom(f ), i.e. it satisfies all propositional atoms and hence (W , R, V ), v | = ψ.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

Relation between U(n) and U⋆(n)

Lemma There exists a positive morphism F : U(n) → U⋆(n), that is onto and for every w ∈ dom(F) we have that F ↾ U(n)w is onto U⋆(n)F(w). Proof. We construct F by induction on the levels of U(n). If w ≺ {w1, . . . , wk}, take A ⊆ F[{w1, . . . , wk}] the set that contains the R⋆-minimal elements of F[{w1, . . . , wk}]. If A is empty then let F(w) to be the element of U⋆(n) with depth 1, with the same color as w. If A = {u} and u has the same color as w then let F(w) = u. Otherwise by the construction of U⋆(n) there a unique v ≺ A (by the induction hypothesis about F) with the same color as w and we let F(w) = v.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

U ⋆(n) witnesses every counterexample

Theorem For any finite rooted intuitionistic n-model M = (M, R, V ) such that for some x ∈ M and p ∈ Prop with x / ∈ V (p), there exists unique w ∈ U⋆(n) and positive morphism of M onto U⋆(n)w. Proof. We know there is a unique such p-morphism to the universal

• model. We take the composition with F. It is still unique since
• therwise if g1, g2 were different positive morphism, since

dom(g1) = dom(g2) = {x ∈ M : ∃p ∈ Prop(x / ∈ V (p))}, we would have two different p-morphisms from dom(g1) to U(n), a contradiction.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

U ⋆(n) witnesses every counterexample

Theorem For every n-formula ϕ ∈ [∨, ∧, →], U⋆(n) | = ϕ if and only if ⊢IPC ϕ. Proof. Follows from previous Lemma.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

de Jongh formulas for U ⋆ (n)

We have that (U⋆(n))+ is (isomorphic to) a generated submodel of U(n), whose domain consist of the elements of U(n) whose only successor of depth 1 satisfies all propositional atoms. Let’s call this generated submodel M. Definition If d(w) = 1 then define ϕ⋆

w =

• prop(w) ∧ (
• notprop(w) →
• notprop(w))

and ψ⋆

w = ϕ⋆ w →

• i∈n

pi.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

de Jongh formulas for U ⋆ (n)

Definition If d(w) > 1 then let w ≺ {w1, . . . , wr} and define ϕ⋆

w =

• prop(w) ∧ (
• newprop(w) ∨
• i≤r

ψ⋆

wi →

• i≤r

ϕ⋆

wi)

and ψ⋆

w = ϕ⋆ w →

• i≤r

ϕ⋆

wi.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) is universal

de Jongh formulas

Proposition For every w ∈ U⋆(n) we have that V ⋆(ϕw) = R⋆(w) V ⋆(ψw) = U⋆(n) \ (R⋆)−1(w) Proof. We can show that for every world w in M, ϕw is top-model equivalent to ϕ⋆

• w. And since ϕ⋆

w is negation free it is satisfied in a

world of (U⋆(n))+ if and only if it is satisfied in the same world in U⋆(n).

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) and H⋆(n)

Basic Relation

We denote the n-Henkin model for the [∨, ∧, →] fragment of IPC with H⋆(n). We write Cn⋆

n(ϕ) = {ψ ∈ [∨, ∧ →] : ψ is an n-formula and ⊢IPC ϕ → ψ}

and we write Th⋆

n(M, w) = {ϕ ∈ [∨, ∧ →] : ϕ is an n-formula and M, w |

= ϕ}. Proposition For any point w ∈ U⋆(n), Th⋆

n(U⋆(n), w) = Cn⋆ n(ϕ⋆ w).

Proof. If IPC ϕ⋆

w → σ, then this is witnessed in some world v of U⋆(n).

We have that v ∈ R⋆(w), hence σ / ∈ Th⋆

n(U⋆(n), w).

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) and H⋆(n)

Basic Relation

Proposition For any w ∈ U⋆(n) we have H⋆(n)Cn⋆(ϕ⋆

w) ∼

= (U⋆(n)w)+. Proof. We have that g : (U⋆(n)w)+ → H⋆(n)Cn⋆(ϕ⋆

w), such that

g(v) = Cn⋆

n(ϕ⋆ v) and the topmost element is mapped to the set of

all negation-free formulas, is the isomorphism. If Γ ⊇ Cn⋆(ϕ⋆

w),

then Γ = Cn⋆(ϕ⋆

v) for wR⋆v, or it contains all propositional atoms:

If there is some v such that ϕv ∈ Γ but for all immediate successors of v, vi ϕ⋆

vi /

∈ Γ (i ∈ n + 1) for σ ∈ Γ we have σ ∧ ϕ⋆

v IPC ϕ⋆ v0 ∨ · · · ∨ ϕ⋆

• vn. Then this is witnessed in U⋆(n),

exactly at v, hence σ ∈ Cn⋆(ϕ⋆

v).

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) and H⋆(n)

Corollaries

Corollary It is the case that Upper(H⋆(n)) ∼ = (U⋆(n))+. Corollary Let M = (M, R, V ) be any n-model and let x ∈ M be such that M, x | = ϕ⋆

w, for some w ∈ U⋆(n). Then either there are a unique

v ∈ U⋆(n) such that wR⋆v, and a positive morphism f from Mx

• nto U⋆(n)v or Mx satisfies all negation-free formulas.

Proof. Define f (y) = v, where Th⋆

n(M, y) = Cn⋆ n(ϕ⋆ v).

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

U⋆(n) and H⋆(n)

Analogue of Jankov’s theorem

Theorem For every descriptive frame G and w ∈ U⋆(n) we have that G ψ⋆

w if and only if there is an n-valuation V on G such that

U⋆(n)w is the image, through a positive morphism, of a generated submodel of (G, V ). Proof. If w ≺ {w1, . . . , wn}, then take the submodel generated by the elements that satisfy ϕ⋆

w but none of the ϕ⋆

• wi. The previous

corollary gives the positive morphism.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Application

Jankov’s theorem for KC

Lemma If F is a descritpive frame with a topmost element, and f : (G, V ) → (F, V ′) is a descriptive positive morphism between models, then f can be extended to a descriptive frame p-morphism. Proof. If f is a total then it is a frame p-morphism. If f is not total then, extend f to f ′ such that every y ∈ dom(G) \ dom(f ), f ′(y) = x0, where x0 is the topmost element of F. To show that it is descriptive we need that f ′−1[Q] is admissible, where Q is admissible in F. But, by the construction of f we have that f ′−1[Q] = f −1[Q] ∪ (dom(G) \ dom(f )), which is admissible by assumption.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

Application

Jankov’s theorem for KC

Theorem (Jankov) For every logic L KC which is complete with respect to a class

• f Kripke frames there exists some negation-free formula σ such

that L ⊢ σ while IPC σ. Proof. Let χ be the formula that L proves. Let F a finite KC frame, a counterexample to χ. We give a valuation V to F such that at every world a propositional atom is not true. Given a L-frame, G, if for any valuation it satisfies the same negation free formulas as (F, V ) then by the previous theorem there is a descriptive positive morphism onto F. This can be extended to a descriptive p-morphism by the above lemma, a contradiction.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC

References

De Jongh, Dick and Yang, Fan Jankov’s theorems for intermediate logics in the setting of universal models, Proceedings of the 8th international tbilisi conference on Logic, language, and computation, 2011. Bezhanishvili, G. and Bezhanishvili, N., An algebraic approach to canonical formulas: intuitionistic case, Rev. Symb. Log, 2009, Cambridge Univ Press Bezhanishvili, N. Lattices of Intermediate and Cylindric Modal Logics, PhD Thesis, Univesity of Amsterdam, The Netherlands, 2006.

Apostolos Tzimoulis and Zhiguang Zhao The Universal Model for the negation-free fragment of IPC